MHF4U
UNIT 2.1: CHARACTERISTICS OF POLYNOMIAL FUNCTIONS
Polynomial Function – A function whose equation is defined by a polynomial in one variable. The general
equation is: 𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + 𝑎𝑛−2 𝑥 𝑛−2 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥1 + 𝑎0
Degree of a Function – the value of the highest exponent of the variable
Leading Coefficient – the coefficient of the term with the highest exponent
Increasing – the graph rises going from left to right along the x-axis
Decreasing – the graph falls going from left to right along the x-axis
Turning Point – occurs where the function changes from increasing to decreasing or vice versa
Local maximum/minimum – point where the function changes from increasing to decreasing/decreasing to
increasing
Local maximum/minimum value – y-coordinate of the local maximum/minimum
Zeroes – a.k.a. roots, x-intercepts
End Behaviour – The behavior of the y-values as x approaches positive infinity and as x approaches
negative infinity
Finite Differences – For a polynomial function of degree n, 𝑛𝜖Ζ+ , the nth differences are equal and have the
same sign (+/-) as the leading coefficient.
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MHF4U
UNIT 2.1: CHARACTERISTICS OF POLYNOMIAL FUNCTIONS
INVESTIGATING POLYNOMIAL FUNCTIONS
Odd Degree Polynomial Functions
Group A
Graph
y  x3
y  x3  x2  4x  4
y  x3  5x2  3x  9
y  x3
y  x3  x2  4x  4
y  x3  5x2  3x  9
End Behaviour
# of Turning Points
# of x-intercepts
Group B
Graph
End Behaviour
# of Turning Points
# of x-intercepts
Compare the graphs in Group A and B. How are they different?
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MHF4U
UNIT 2.1: CHARACTERISTICS OF POLYNOMIAL FUNCTIONS
Even Degree Polynomial Functions
Group A
Graph
y  x4
y  x4  x3  6x2  4x  8
y  x4  3x3  3x2  11x  4
y  x4
y  x4  x3  6x2  4x  8 y  x4  3x3  3x2  11x  4
End Behaviour
# of Turning
Points
# of x-intercepts
Group B
Graph
End Behavior
# of Turning
Points
# of x-intercepts
Compare the graphs in Group A and Group B. How are they different?
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MHF4U
UNIT 2.1: CHARACTERISTICS OF POLYNOMIAL FUNCTIONS
SUMMARY
Odd Degree Functions
Type
General Equation
Degree of
Polynomial
Maximum
Number of
Roots
Max Number
of Turning
Points
Sketch
Linear
Cubic
Quintic
A polynomial of degree 7 is called a septic.
9 is called a nonic.
End Behaviour
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MHF4U
UNIT 2.1: CHARACTERISTICS OF POLYNOMIAL FUNCTIONS
Even Degree Functions
Type
General Equation
Degree of
Polynomial
Maximum
Number of
Roots
Max Number
of Turning
Points
Sketch
Quadratic
Quartic
Hexic
(or sextic)
A polynomial of degree 8 is called an octic.
degree 10 is decic.
End Behaviour
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